For any two vector fields \(\xi ,\lambda \in T_{\left[\bar{{\cal X}}\right]}\overline{{\cal M}}_{\mathbf {r}}/{\cal G}\) , the horizontal lift of the Riemannian connection is given by  [1]}
\(\overline{\nabla _{\xi _{[\bar{{\cal X}}]}}\lambda } = {\cal P}_{\bar{{\cal X}}}^{{\cal H}}\left(\nabla _{\bar{\xi }_{\bar{{\cal X}}}}\bar{\lambda } \right),\) 
